# Can a deformable object "swim" in curved space-time? [duplicate]

Possible Duplicate:
Swimming in Spacetime - apparent conserved quantity violation

It is well known that a deformable object can perform a finite rotation in space by performing deformations - without violating the law of conservation of angular momentum since the moment of inertia can be changed by the deformations of the object, see e.g. this Phys.SE question.

It is also well known that in flat space-time, it is not possible for a deformable object to displace it's center of gravity by performing deformations, see e.g. this Phys.SE question.

However in curved space-time can a deformable object swim through space by performing deformations?

Yes, you can swim through space, but only if space is curved - in the vicinity of a gravitating body (which creates curvature of space-time) it is possible for an isolated body to move by only executing internal motions of parts of the body. The reason this is possible is that the center of gravity of an object is not well defined in a curved space-time. Therefore a set of deformations of the object can result in a net displacement of the not well defined center of mass

A deformable object with no angular momentum can perform a rotation without violating the law of conservation of angular momentum. A simple way of imagining this is to think of 4 weights (red) on 2 tracks (black) that are connected by a motor (green) that can rotate the two rods around their centers of mass as seen as this picture:

Starting with this object stationary in space and with no angular momentum, move two weights closer to the pivot point. Then rotate the rods with the green motor - the rod with the weight far apart will rotate through a smaller angle than the weights that are closer together since the moment of inertia of the two rods are different. Then reverse the positions of the weights on the two rods and reverse the angle of rotation of the green motor. When you finish this maneuver there will be an overall change of the angle of the object in space. However at the end of the maneuver the center of mass of the object will be in the same position and the total angular momentum will still be zero - thus the angular momentum is still conserved. You can actually do this yourself using your arms and legs and a swiveling desk chair.

In the same way that angular momentum was conserved in the rotation case, linear momentum will be conserved for the displacement in curved space case - if the object was stationary before the maneuver it will be stationary after the maneuver and thus there is no violation of linear momentum conservation. Even though there is a displacement of the center of mass caused by the maneuver there is no continuing motion of the center of mass. However this kind of displacement maneuver is only possible in curved space-time - in flat space-time this kind of displacement is impossible since the center of mass is well defined whereas the center of masss is not well defined in curved space-time.

In "ordinary" curved space-time, such as the curvature caused by the Earth, the effects are very small, but they are non-zero. For example for a meter sized object performing meter sized movement sequences in the vicinity of the Earth, the distance the center of mass would move is only $10^{-34}$ meters for each sequence of moves (this distance is only about 6 Planck lengths!).

A blog that gives a very readable and perhaps understandable description of the effect is here: http://www.science20.com/hammock_physicist/swimming_through_empty_space

The blog explains:

...[the] key to this swimming in empty space is the fact that the concept of center-of-mass is ill-defined in non-Euclidean space. Non-Euclidean space swimming is geometric in nature and entirely determined by the sequence of shapes assumed. In several ways this swimming is similar to the mechanism by which a cat falling upside-down rotates itself during free fall. Physicists refer to geometric phases to describe these effects...

And gives a 2 dimensional example of swimming:

Imagine a two-dimensional three-legged creature moving frictionlessly over the surface of a sphere.

Let's say this creature is positioned at the equator with one leg pointing east and two legs pointing along the lines of longitude towards the north and south poles. The swim stroke consists of four moves. First the tripodal creature extends its two longitudinal legs, and subsequently it extends the eastern leg. To complete the stroke, it retracts the longitudinal legs and finaly retracts the eastern leg. As a result of each such 'swimstroke' the creature moves a wee bit westward.

Why is this?

Key is that when the eastern leg extends, the longitudinal legs are extended away from the equator, whilst when the eastern leg retracts, the they are closer to the equator. If the creature keeps its longitudinal legs all the time oriented along the sphere's lines of longitude, the backreaction to the eastern leg extending translates into a smaller movement at the tip of the longitudinal legs, and a larger movement at the base of the longitudinal legs located at the equator.

The reverse is true for the backreaction to the eastern leg retracting. As a result, the westward movement of the base of the longitudinal legs along the equator is larger than its eastward movement.

A video that shows how 3 connected weights can swim on a two dimensional sphere is available here: http://www.iop.org/EJ/mmedia/1367-2630/8/5/068/movie1.avi . Here is a frame from that video:

This two dimensional example also shows the difficulty with defining the center of mass in curved space. For example, in our 3 dimensional space where the 2 dimensional sphere is embedded, we can see that the center of mass of these three masses will be inside the sphere, yet we are assuming the 3 mass object is only in a 2 dimensional curved space. Since the 2-D center of mass must be on the surface of the sphere, the best we can do is to project the 3-D center of mass onto the surface of the sphere. In particular, when the two masses are closer to the two poles, the center of mass is deeper inside the sphere. So when the 3rd equator mass moves while the center of mass is deeper inside the sphere, the projected center of mass on the surface of the sphere will move a larger distance than when the other two weights are closer to the equator and the center of mass is closer to the surface. That is how the net displacement of the 2-D center of mass is achieved.

A link to the very technical original research article is: http://dspace.mit.edu/bitstream/handle/1721.1/6706/AIM-2002-017.pdf?sequence=2